and, commute on If. (i) is continuous and weakly -contractive or.
A function is said to be rd-continuous (denoted by if. (i) is continuous at every right-dense point, (i) exists and is finite at every left-dense point. . is continuous at every right-dense point, exists and is finite at every left-dense point.
Remark 3.5 (i) It is easy to see that if M i is continuous on C, then M i is hemicontinuous and bounded on any line segment of C. (ii) It is easy to see that Theorem 3.10 is true if for each i ∈ N, M i : C → H is a κ i -inverse strongly monotone mapping or is a continuous strongly monotone mapping.
If, then one has (i) is continuous for all,, for all ; (ii), for all,. . is continuous for all,, for all ;, for all,.
A function ψ : I → ℝ is exponentially convex on I if it is continuous and ∑ i, j = 1 n ξ i ξ j ψ ( x i + x j ) ≥ 0. ∀n ∈ ℕ and all choices ξ i ∈ R; i = 1,..., n such that x i + x j ∈ I; 1 ≤ i, j ≤ n.
A function h : I → ℝ is exponentially convex on I if it is continuous and.
A function (psi Irightarrowmathbb{R}) is exponentially convex on I if it is continuous and sum_{i,j=1}^{n} xi_{i} xi_{j} psi (x_{i}+x_{j} )geq0 for all (nin mathbb{N}) and all choices (xi_{i}inmathbb{R}), (i=1,dots,n), such that (x_{i}+x_{j}in I), (1leq i,jleq n).
Definition 1. [[3], p. 373] A function f : (a, b) → ℝ is exponentially convex if it is continuous and ∑ i, j = 1 n ξ i ξ j f ( x i + x j ) ≥ 0 (1).
Definition 2.1 A function γ : R + → R + is a (i) K-function if it is continuous, strictly increasing and γ ( 0 ) = 0, and (ii) K ∞ -function if it is a K-function and also γ ( r ) → ∞ as r → ∞. .
A mapping (T:mathcal {D}subseteq Xrightarrow Y) is said to be: (i) ws-compact if it is continuous and maps relatively weakly compact sets of (mathcal{D}) into relatively strongly compact ones of Y; (ii) ww-compact if it is continuous and maps relatively weakly compact sets of (mathcal{D}) into relatively weakly compact ones of Y. .
